L(s) = 1 | − i·2-s + (1.92 − 1.11i)3-s − 4-s + (0.628 − 2.14i)5-s + (−1.11 − 1.92i)6-s + (1.64 + 0.950i)7-s + i·8-s + (0.966 − 1.67i)9-s + (−2.14 − 0.628i)10-s + 2.76·11-s + (−1.92 + 1.11i)12-s + (−1.72 − 0.996i)13-s + (0.950 − 1.64i)14-s + (−1.17 − 4.82i)15-s + 16-s + (2.11 + 1.22i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (1.11 − 0.641i)3-s − 0.5·4-s + (0.281 − 0.959i)5-s + (−0.453 − 0.785i)6-s + (0.622 + 0.359i)7-s + 0.353i·8-s + (0.322 − 0.557i)9-s + (−0.678 − 0.198i)10-s + 0.834·11-s + (−0.555 + 0.320i)12-s + (−0.478 − 0.276i)13-s + (0.254 − 0.440i)14-s + (−0.303 − 1.24i)15-s + 0.250·16-s + (0.512 + 0.295i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23463 - 1.62297i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23463 - 1.62297i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (-0.628 + 2.14i)T \) |
| 43 | \( 1 + (-2.06 - 6.22i)T \) |
good | 3 | \( 1 + (-1.92 + 1.11i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.64 - 0.950i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 2.76T + 11T^{2} \) |
| 13 | \( 1 + (1.72 + 0.996i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.11 - 1.22i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.26 + 2.18i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.86 - 4.54i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.47 - 2.56i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.748 + 1.29i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.00 + 2.88i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 47 | \( 1 - 8.63iT - 47T^{2} \) |
| 53 | \( 1 + (7.16 - 4.13i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 0.157T + 59T^{2} \) |
| 61 | \( 1 + (-2.76 + 4.79i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.59 + 2.65i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.80 + 13.5i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.08 - 1.78i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.13 - 1.96i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.45 + 1.99i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.70 + 6.42i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.95iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.04058914084855039265540740569, −9.596624439975345006296012626832, −9.220246792783199928034306357541, −8.158860001513702988049700902847, −7.74373481705025341023000440587, −6.04728322546472280550132609797, −4.88278931040458976285184311305, −3.71449550266972193578088948501, −2.29789936430705659549859152683, −1.42454928144749991833196038553,
2.23249633956771831309296309689, 3.63766609203039526360780036854, 4.37863383820973761955618559347, 5.88301291651853020164223667657, 6.87965005797934154419119235089, 7.85746643093408872164612253308, 8.558617431228489473385429000204, 9.725316432358364197782712598679, 10.04820012787106199831798184656, 11.28149600611859097019117413880